Let A,B ε Mnxn(C)
Prove that if B is invertible, then there exists a scalar cεC s.t. A + cB is not invertible. HINT:
examine det (A+cB)
The Attempt at a Solution
I know that the det(A+cB)=0 since a non invertible matrix has det=0
I know B is invertible so I multiply the right side by B^-1 which gives det(AB^-1 + cIn)=0 where In is the nxn identity matrix. There is a theorem that says that a scalar is an eigenvalue of A if and only if det(A-cIn)=0.
can I choose c to be -c? does that show that A +cB is not invertible? Help!